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12 — Doob’s Upcrossing Lemma and Martingale Convergence

1. Convex Transformations of Martingales

Let ${X_n, \mathcal{F}n}{n\ge1}$ be a martingale.
Let $\varphi:\mathbb{R}\to\mathbb{R}$ be a convex function with
\(E|\varphi(X_n)| < \infty,\qquad n\ge1.\)

Theorem A

Then the process
\(\{\varphi(X_n), \mathcal{F}_n\}_{n\ge1}\)
is a submartingale.
(If $\varphi$ is concave, it becomes a supermartingale.)

Proof sketch:
Using Jensen’s inequality, \(E[\varphi(X_{n+1}) \mid \mathcal{F}_n] \ge \varphi(E[X_{n+1}\mid \mathcal{F}_n]) = \varphi(X_n).\)

Example

$\varphi(x)=x^2$ is convex, so ${X_n^2}$ is a submartingale.

2. Convex + Increasing Functions Preserve Submartingales

If ${X_n}$ is a submartingale and $\varphi$ is convex and increasing,
then ${\varphi(X_n)}$ is also a submartingale.

3. Powers of Martingales

For $p\ge1$:
If ${X_n}$ is a martingale then $|X_n|^p$ is a submartingale.

4. Optional Stopping: Last Time and Stopping Times

If ${X_n,\mathcal{F}_n}$ is a MG (or subMG or superMG) and $T$ is a stopping time,
then
\(\{X_{n\wedge T}, \mathcal{F}_n\}\) is again a MG (or subMG, superMG).

This is the “we can stop and still have a MG” rule.

5. Doob’s Upcrossing Framework

Fix real numbers $a<b$.
Define stopping times: \(\begin{aligned} T_0 &= 1, \\ T_{2k-1} &= \inf\{n > T_{2k-2} : X_n \le a\},\\ T_{2k} &= \inf\{n > T_{2k-1} : X_n \ge b\}. \end{aligned}\)

Define the number of completed upcrossings of $(a,b)$ by time $n$: \(U_n^{a,b}.\)

6. Doob’s Upcrossing Lemma

For a submartingale ${X_n}$,

\[(b-a) E[U_n^{a,b}] \;\le\; E[(X_n-a)^+] - E[(X_0-a)^+].\]

Since $(x+o)^+ \le c^+ + d^+$, we get the bound: \((b-a)E[U_n^{a,b}] \;\le\; E[X_n^+] + |a|.\)

This shows $U_n^{a,b}$ is finite almost surely when $\sup_n E[X_n^+] < \infty$.

7. Martingale Convergence Theorem (MCT)

Let ${X_n,\mathcal{F}_n}$ be a submartingale (or supermartingale).
Assume
\(\sup_{n} E[X_n^+] < \infty.\)

Then:

  1. Almost sure convergence:
    \(X_n \xrightarrow{\text{a.s.}} X.\)

  2. $L^1$–integrability of the limit:
    \(E|X| < \infty.\)

Remark

For submartingales,
\(\sup_n E[X_n^+] < \infty \quad\Longleftrightarrow\quad \sup_n E|X_n| < \infty.\)

8. Example 1 — Stopped Simple Symmetric Random Walk

Let $S_n = \sum_{k=1}^n \varepsilon_k$, where $\varepsilon_k=\pm1$ iid.
Let
\(T = \inf\{n : S_n = 1\}.\)

Then ${S_{T\wedge n}}$ is a submartingale,
and
\(\sup_n E[(S_{T\wedge n})^+] \le 1.\)

Thus by MGCT: \(S_{T\wedge n} \to 1 \quad \text{a.s.}\)

But it is not uniformly integrable, since
\(E[S_{T\wedge n}] = 0,\qquad E[1]=1.\)

9. Example 2 — Exponential Martingale for Normal Increments

Let ${Z_k}$ be iid $N(0,1)$ and
\(S_n = \sum_{k=1}^n Z_k,\qquad Y_n = e^{S_n - n/2}.\)

Then ${Y_n}$ is a martingale.

Since $Y_n \ge 0$ and $E(Y_n)=1$,
by MCT: \(Y_n \to Y \quad \text{a.s.}\)

But: \(Y_n = e^{n\left(\frac{S_n}{n} - \frac12\right)}, \qquad \frac{S_n}{n} \to 0 \quad \text{a.s.}\)

Thus \(Y_n \to 0 \quad\text{a.s.}\)

and the martingale is not uniformly integrable since
\(E(Y_n)=1 \not\to 0.\)

Summary

  • Convex (and convex + increasing) transforms of martingales produce submartingales.
  • Optional stopping preserves MG/subMG/superMG when stopping times are used.
  • Doob’s Upcrossing Lemma is the key step behind the Martingale Convergence Theorem.
  • MCT: bounded positive expectations imply almost sure convergence.
  • Examples illustrate failure of uniform integrability despite a.s. convergence.
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